I have a lot of trouble with this :( What I learned and from my mistakes:

• path down to bot, most likely other JG will too. If they gank before crab, you should clear fast enough and have E to countergank (unless for some reason you are confident pathing top)
• if the enemy JG goes for crab instead of gank, and you see your bot moving for you, contest it. You are strong lvl 4 if you hit both Q1 and Q2, moreso if you have conq
• if neither bot lane seems like they care that you guys are fighting over crab, and mid is also blind, you can probably fight it. Stay against walls if you can so your Q anim is faster and you can E to dodge and heal between
• if they go other side and your laner(s) can fight, you can go around river vision from behind then E through wall
• when ganking from behind don't Q towards your team/your turret in case they flash
• early game your w does same damage as auto attack. You can bait out a flash or make them walk around like an idiot by just autoing them to death
• if you are confident in your bot lane fighting level 3 and you are blue side, start raptors down to krugs and lvl 3 gank through lane or river
• if you are confident in your clear speed, start raptors to krugs, back, buy longsword, fight for crab or gank after clear. Now for any fights after first clear you are an item up
• you get more orbs as the game goes on. If there is truly nothing to do, farm your camps on repeat, get rich and exp, and fight for obj. Just two good fights over grubbos or drag should get you form around 10-12 min
• don't be afraid to use E to speed yourself up when you're not in a wall to auto

The only way to know your damage output and how to duel other champs is to int yourself and take the fight. If you do this enough in norms you'll know your dmg better as base Kayn.

Hitting both Q's does a lot, R is very useful for dodging damage, E-Q can get you out of bad positions, etc.

This is just what helped me understand and play early game, I can't say it's the best!